Non-Hermitian Phase Transition

• Non-Hermitian phase transition is a structural change in the spectrum, eigenstate localization, and dynamics triggered by tuning non-Hermitian parameters.
• Key features include the emergence of exceptional points, abrupt jumps in topological invariants, and a breakdown of conventional bulk-boundary correspondence.
• Experimental realizations in photonic, cold atom, and electronic systems highlight its potential for dynamic control, sensing, and advancing quantum technologies.

A non-Hermitian phase transition (NHPT) refers to a qualitative structural change in the spectrum, eigenstate localization, or dynamical properties of a quantum or classical system described by a non-Hermitian Hamiltonian, typically triggered by tuning a control parameter such as disorder strength, coupling asymmetry, dissipation, or complex potential. The transition is intrinsically connected to the non-Hermitian nature: eigenvalues may become complex, exceptional points can arise where eigenvalues/eigenvectors coalesce, and topological invariants based on complex spectral geometry display quantized jumps. In contrast to Hermitian transitions, NHPTs often occur in open, non-equilibrium, or dissipative environments, and may feature breakdowns of bulk-boundary correspondence, non-Hermitian skin effects, or reentrant topological behavior.

1. Canonical Models and Spectral Characterization

A paradigmatic setting is the non-Hermitian Aubry-André-Harper (AAH) quasicrystal, where the single-particle Hamiltonian on a 1D lattice is

$$[H(\theta,h)\psi]_n = J\,(\psi_{n+1} + \psi_{n-1}) + V\,\cos(2\pi\alpha n + \theta + i h)\,\psi_n,$$

with $J$ hopping amplitude, $V$ real modulation, $\alpha$ irrational wave vector (quasiperiodicity), $\theta$ real, $h$ imaginary phase (non-Hermiticity, $\mathcal{PT}$ symmetry for $\theta=0$). In the thermodynamic limit ($L \to \infty$), the transition between extended (metallic, real spectrum) and localized (insulator, complex spectrum) states occurs when

$h_c = \ln\left(\frac{2J}{V}\right).$

For $h < h_c$, all eigenenergies are real and states are extended; for $h > h_c$, the spectrum becomes complex and all states are exponentially localized. This transition is topological: the spectral winding number

$$w(h) = \lim_{L \to \infty} \frac{1}{2\pi i} \int_0^{2\pi} d\theta\, \partial_\theta \ln \det [H(\theta/L, h) - E_B]$$

changes from $w=0$ (metallic) to $w=-1$ (insulator) exactly at $h_c$ (Longhi, 2019).

2. Topological Invariants and Geometric Criteria

Non-Hermitian topological phase transitions are distinguished by the behavior of spectral winding numbers, biorthogonal Chern/Z$_2$ indices, and explicit geometric criteria. For example, in one and two dimensions, the length $L(\lambda)$ of the boundary of the bulk band in the complex energy plane

$$L(\lambda) = \oint_\mathcal{C} |dE| = \int_{-\pi}^{\pi} \sqrt{\left(\partial_k E_R\right)^2 + \left(\partial_k E_I\right)^2}\; dk$$

undergoes nonanalytic behavior at the phase boundary. In 1D, the derivative $\partial_\lambda L$ diverges or jumps when the winding number of the complex Bloch band around the origin changes (Fan et al., 2023). In 2D, $L$ itself jumps discontinuously at gapped-gapless transitions.

Phase transitions in non-Hermitian quantum spin Hall insulators can be induced solely by varying non-Hermitian terms, and characterized by the biorthogonal Wilson loop Z$_2$ invariant or biorthogonal spin Chern number, which correctly track the transitions between trivial and non-trivial phases and the appearance of exceptional edge arcs (Hou et al., 2019).

3. Dynamical and Critical Behavior

Unlike Hermitian cases, non-Hermitian phase transitions can present discontinuous dynamical features at finite system sizes due to branch-point singularities (exceptional points). For instance, wavepacket spreading in a non-Hermitian AAH model can exhibit jumps in the quantum diffusion exponent $\delta$ and ballistic velocity $v(V)$ across the localization transition, with $v(V)$ remaining strictly positive up to the transition—a disorder-enhanced transport phenomenon unique to non-Hermitian systems (Longhi, 2021).

Universal critical behaviors are governed by the coalescence order $p$ at exceptional points: observables and entanglement measures exhibit scaling exponents $(1-1/p), 2(1-1/p), \ldots, n(1-1/p), \ldots$ near the transition (Wei et al., 2016). Order parameters and susceptibilities may display true nonanalytic power laws even at finite size, in sharp contrast to Hermitian transitions, and enter new universality classes (He et al., 2022).

For systems far from equilibrium, the NHPT may manifest in dynamical response rather than static order. For instance, coupled resonators subjected to different bath temperatures show a Landau-like transition in the frequency spectrum of energy flow, where spectral peak splitting plays the role of the order parameter. The critical exponent for fluctuations is non-universal, depending only on the temperature ratio, and the transition occurs without any exceptional point in the Hamiltonian (Sergeev et al., 2021).

4. Exceptional Points, Re-entrant and Unusual Topological Transitions

Exceptional points (EPs), where multiple eigenvalues and eigenvectors coalesce, are central to non-Hermitian transitions. EP crossing leads to square-root scaling in relaxation rates, as observed experimentally in bulk EuO: time-resolved reflection shows a transition from bi-exponential real decay to single-mode complex decay, with $\gamma''(T) \propto \sqrt{T-T^*}$ above the EP (Li et al., 2024).

Novel re-entrant behavior is possible. In generalized non-Hermitian AAH models, delocalization–localization–delocalization transitions arise with increasing non-Hermiticity parameter $h$, each regime assigned a spectral winding number. Intermediate phases host both real (extended) and imaginary (localized) states, separated by mobility edges in the complex plane, and feature a distinct set of winding numbers per spectral loop (Padhan et al., 2023). Likewise, in non-Hermitian SSH models, transitions between two non-trivial insulating phases can be mediated by a robust non-trivial semi-metallic phase, with distinct edge-state character and complex energy spectra (Niveth et al., 2024).

5. Non-Hermitian Skin Effect, Bulk-Boundary Breakdown, and Localization

Non-Hermiticity often results in the failure of conventional bulk-boundary correspondence. In a non-Hermitian AAH quasicrystal, the winding number does not predict the number or chirality of edge modes, and the macroscopic non-Hermitian skin effect is absent under open boundary conditions; edge states can exist even in the metallic regime ($w=0$), and the PT phase is fragile to boundary introduction (Longhi, 2019).

In translation-invariant non-Hermitian tight-binding models with asymmetric couplings (interpreted as imaginary gauge fields), phase transitions from real to complex spectrum are immediate, but under open boundary conditions, the spectrum remains real and all right-eigenstates localize at one edge (skin effect). Biorthogonal norms restore extended character, indicating the effect is basis-dependent (Wang et al., 2019).

Disordered non-Hermitian systems (e.g., Hatano-Nelson chain) display dynamical phase transitions between localization, unidirectional amplification, and distinct propagating phases, with critical exponents differing from Hermitian Anderson transitions—e.g., $\nu=1/2$ for metal-metal transitions (Spring et al., 2023).

6. Many-body Quantum Phase Transitions and Entanglement

Non-Hermitian extensions of many-body paradigms (XY, Lipkin-Meshkov-Glick, quantum contact process) support both Hermitian-like and novel NHPTs. In the non-Hermitian XY chains, ground-state correlations and entanglement scaling can replicate Hermitian universality class features if the biorthogonal framework is used; critical exponents for correlation length ($\nu=1$) and entanglement entropy central charge ($c=1$) are preserved (Wang et al., 15 Nov 2025, Liu et al., 23 Jan 2025).

In the non-Hermitian LMG model, phase transitions occur with maximal multiparticle entanglement and spin squeezing at the exceptional point, outperforming Hermitian analogs. Finite-size transitions are sharp (full N-particle entanglement), associated with a coalescence of eigenvectors and nonanalytic observables (Lee et al., 2014). Non-Hermitian quantum contact process transitions show uniquely continuous, infinitely singular scaling (e.g., $\beta=1/2$, $\gamma=3/2$) and dissipation-induced universality (He et al., 2022).

7. Experimental Realizations and Application Directions

NHPTs have been realized in ultrafast time-resolved experiments (EuO, bright-dark exciton relaxation), cold atoms under non-Hermitian dynamics (heralded magnetism, XY chains), photonic quantum walks (synthetic superlattices), and mode-locked lasers (spectral broadening as a signature of the metal-insulator transition) (Li et al., 2024, Lee et al., 2014, Longhi, 2023, Longhi, 2019). Whenever open system platforms combine gain/loss, non-reciprocal couplings, and tunable control parameters, non-Hermitian phase transitions—often governed by geometry in the complex energy plane—can be invoked to control many-body dynamics, sensing, topological amplification, and quantum metrology.

Summary Table: Principal NHPT Scenarios

Model Class	Transition Type	Topological Quantity / Observable
AAH Quasicrystals	Metal–Insulator	Spectral winding number ($w$)
XY, LMG Chains	Magnetic/Disorder	Entanglement, Fisher info, $\nu$, $c$
SSH, Chern Models	Trivial–Topological	Biorthogonal Zak, Chern, Z$_2$ invariants
Bulk EuO	Dynamical (EP)	Relaxation rates, $\sqrt{T-T^*}$ scaling
Superlattices	Localization (Disorder)	IPR, spectral gap closure
Coupled Oscillators	Non-equilibrium	Frequency splitting, Landau order parameter

Transitions are often marked by winding-number jumps, geometric nonanalyticities, exceptional-point coalescence, or singular dynamical observables. These phenomena are robust features of open quantum system behavior in the non-Hermitian regime, with applications in metrology, dynamical control, and the construction of new classes of topological materials.